Difference between revisions of "Non-ideal relay"
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| − | + | The [[Hysteresis|hysteresis]] non-linearity denoted by $ {\mathcal R} ( \alpha, \beta ) $, | |
| + | with thresholds $ \alpha $ | ||
| + | and $ \beta $, | ||
| + | and defined for a continuous input $ u ( t ) $, | ||
| + | $ t \geq t _ {0} $, | ||
| + | and an initial state $ r _ {0} \in \{ 0,1 \} $ | ||
| + | by the formulas (see Fig.a1.) | ||
| + | |||
| + | $$ | ||
| + | {\mathcal R} ( r _ {0} ; \alpha, \beta ) u ( t ) = \left \{ | ||
| + | |||
| + | \begin{array}{ll} | ||
| + | r _ {0} &\textrm{ if } \alpha < u ( s ) < \beta, t _ {0} \leq s \leq t, \\ | ||
| + | 0 &\textrm{ if either } u ( t ) \leq \beta \textrm{ or } \\ | ||
| + | {} & u ( t ) \in ( \beta, \alpha ) \textrm{ and } u ( \tau ) = \beta, \\ | ||
| + | 1 &\textrm{ if either } u ( t ) \geq \alpha \textrm{ or } \\ | ||
| + | {} & u ( t ) \in ( \beta, \alpha ) \textrm{ and } u ( \tau ) = \alpha, \\ | ||
| + | \end{array} | ||
| + | \right . | ||
| + | $$ | ||
| + | |||
| + | where $ \tau = \sup \{ s : {s \leq t, u ( s ) = \beta \textrm{ or } u ( s ) = \alpha } \} $, | ||
| + | that is, $ \tau $ | ||
| + | denotes the last switching moment. The input–output operators $ {\mathcal R} ( r _ {0} ; \alpha, \beta ) $ | ||
| + | are discontinuous in the usual function spaces. These operators are monotone in a natural sense, which allows one to use the powerful methods of the theory of semi-ordered spaces in the analysis of systems with non-ideal relays. | ||
<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/n110060a.gif" /> | <img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/n110060a.gif" /> | ||
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Non-ideal relay | Non-ideal relay | ||
| − | The Preisach–Giltay model of ferromagnetic hysteresis is described as the spectral decomposition in a continual system of non-ideal relays in the following way. Let | + | The Preisach–Giltay model of ferromagnetic hysteresis is described as the spectral decomposition in a continual system of non-ideal relays in the following way. Let $ \mu ( \alpha, \beta ) $ |
| + | be a finite [[Borel measure|Borel measure]] in the half-plane $ \Pi = \{ {( \alpha, \beta ) } : {\alpha > \beta } \} $. | ||
| + | The input–output operators of the Preisach–Giltay hysteresis non-linearity at a given continuous input $ u ( t ) $, | ||
| + | $ t \geq t _ {0} $, | ||
| + | and initial state $ S ( t _ {0} ) $ | ||
| + | is defined by the formula | ||
| − | + | $$ | |
| + | x ( t ) = \int\limits { {\mathcal R} ( r _ {0} ( \alpha, \beta ) ; \alpha, \beta ) u ( t ) } {d \mu ( \alpha, \beta ) } , | ||
| + | $$ | ||
| − | where the measurable function | + | where the measurable function $ r _ {0} ( \alpha, \beta ) $ |
| + | describes the internal state of the non-linearity at the initial moment $ t = t _ {0} $. | ||
| + | In contrast to the individual non-ideal relay, the operators of a Preisach–Giltay non-linearity are continuous in the space of continuous functions, provided that the measure $ \mu ( \alpha, \beta ) $ | ||
| + | is absolutely continuous with respect to the [[Lebesgue measure|Lebesgue measure]] (cf. [[Absolute continuity|Absolute continuity]]). For detailed properties of Preisach–Giltay hysteresis and further generalizations see [[#References|[a1]]], [[#References|[a2]]] and the references therein. | ||
See also [[Hysteresis|Hysteresis]]. | See also [[Hysteresis|Hysteresis]]. | ||
Latest revision as of 14:54, 7 June 2020
The hysteresis non-linearity denoted by $ {\mathcal R} ( \alpha, \beta ) $,
with thresholds $ \alpha $
and $ \beta $,
and defined for a continuous input $ u ( t ) $,
$ t \geq t _ {0} $,
and an initial state $ r _ {0} \in \{ 0,1 \} $
by the formulas (see Fig.a1.)
$$ {\mathcal R} ( r _ {0} ; \alpha, \beta ) u ( t ) = \left \{ \begin{array}{ll} r _ {0} &\textrm{ if } \alpha < u ( s ) < \beta, t _ {0} \leq s \leq t, \\ 0 &\textrm{ if either } u ( t ) \leq \beta \textrm{ or } \\ {} & u ( t ) \in ( \beta, \alpha ) \textrm{ and } u ( \tau ) = \beta, \\ 1 &\textrm{ if either } u ( t ) \geq \alpha \textrm{ or } \\ {} & u ( t ) \in ( \beta, \alpha ) \textrm{ and } u ( \tau ) = \alpha, \\ \end{array} \right . $$
where $ \tau = \sup \{ s : {s \leq t, u ( s ) = \beta \textrm{ or } u ( s ) = \alpha } \} $, that is, $ \tau $ denotes the last switching moment. The input–output operators $ {\mathcal R} ( r _ {0} ; \alpha, \beta ) $ are discontinuous in the usual function spaces. These operators are monotone in a natural sense, which allows one to use the powerful methods of the theory of semi-ordered spaces in the analysis of systems with non-ideal relays.
Figure: n110060a
Non-ideal relay
The Preisach–Giltay model of ferromagnetic hysteresis is described as the spectral decomposition in a continual system of non-ideal relays in the following way. Let $ \mu ( \alpha, \beta ) $ be a finite Borel measure in the half-plane $ \Pi = \{ {( \alpha, \beta ) } : {\alpha > \beta } \} $. The input–output operators of the Preisach–Giltay hysteresis non-linearity at a given continuous input $ u ( t ) $, $ t \geq t _ {0} $, and initial state $ S ( t _ {0} ) $ is defined by the formula
$$ x ( t ) = \int\limits { {\mathcal R} ( r _ {0} ( \alpha, \beta ) ; \alpha, \beta ) u ( t ) } {d \mu ( \alpha, \beta ) } , $$
where the measurable function $ r _ {0} ( \alpha, \beta ) $ describes the internal state of the non-linearity at the initial moment $ t = t _ {0} $. In contrast to the individual non-ideal relay, the operators of a Preisach–Giltay non-linearity are continuous in the space of continuous functions, provided that the measure $ \mu ( \alpha, \beta ) $ is absolutely continuous with respect to the Lebesgue measure (cf. Absolute continuity). For detailed properties of Preisach–Giltay hysteresis and further generalizations see [a1], [a2] and the references therein.
See also Hysteresis.
References
| [a1] | M.A. Krasnosel'skii, A.V. Pokrovskii, "Systems with hysteresis" , Springer (1989) (In Russian) |
| [a2] | I.D. Mayergoyz, "Mathematical models of hysteresis" , Springer (1991) |
Non-ideal relay. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Non-ideal_relay&oldid=49491